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A Riesz representation theorem for cone-valued functions. (English) Zbl 0983.46033

The author defines Borel measures on a locally compact Hausdorff space and measurable functions with values in linear functionals on a locally convex cone. In this context the concepts of measure theory for real-valued functions can be applied. He also defines integrals for cone valued functions. A theorem of Riesz representation type is proved. It shows that continuous linear functionals on certain spaces of continuous cone valued functions endowed with a certain inductive limit topology can be represented by such integrals.

MSC:

46E40 Spaces of vector- and operator-valued functions
46G12 Measures and integration on abstract linear spaces
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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